partially_ordered_groups - MathStructures

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=====Partially ordered groups=====

Abbreviation: **PoGrp**

====Definition====
A \emph{partially ordered group} is a structure $\mathbf{G}=\langle G,\cdot,^{-1},1,\le\rangle$ such that

$\langle G,\cdot,^{-1},1\rangle$ is a [[group]]

$\langle G,\le\rangle$ is a [[partially ordered set]]

$\cdot$ is \emph{orderpreserving}:  $x\le y\Longrightarrow wxz\le wyz$

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be partially ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an orderpreserving homomorphism: 
$h(x \cdot y)=h(x) \cdot h(y)$,
$x\le y\Longrightarrow h(x)\le h(y)$

====Examples====
Example 1: The integers, the rationals and the reals with the usual order.

====Basic results====

Any [[group]] is a partially ordered group with equality as partial order.

Any finite partially ordered group has only the equality relation as partial order.


====Properties====

^[[Classtype]]                        |quasivariety  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &1\\
  f(3)= &1\\
  f(4)= &2\\
  f(5)= &1\\
\end{array}$     
$\begin{array}{lr}
  f(6)= &2\\
  f(7)= &1\\
  f(8)= &5\\
  f(9)= &2\\
  f(10)= &2\\
\end{array}$


====Subclasses====
[[Abelian partially ordered groups]]

[[Lattice-ordered groups]] expanded type


====Superclasses====
[[Partially ordered monoids]] reduced type


====References====

[(Lastname19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

Trace: • commutative_residuated_partially_ordered_monoids • universal_classes • shells • categories • hausdorff_spaces • abelian_lattice-ordered_groups • cancellative_residuated_lattices • commutative_residuated_partially_ordered_semigroups • partial_monoids • quasitrivial_groupoids